Interpretability of various extensions of arithmetic
type: Item
instance of: scholarly article
title: Interpretability of various extensions of arithmetic
zbMATH Open document ID: 0632.03044
DOI: 10.1007/BF01159698
author: S. V. Goryachev
published in: Mathematical Notes
publication date: 1986-01-01T00:00:00Z
review text: The paper proves some nice theorems about interpretability of extensions of arithmetic PA. The main result: let \(PA_ 0=PA\) and \(PA_{n+1}=PA_ n+Consis PA_ n\), then PA \(+\) arbitrary m examples of the local reflection principle \(\Pr_{PA}(\ulcorner \Phi \urcorner)\to \Phi\) is interpretable in the theory \(PA_ n\) iff \(m\leq n\). It follows immediately that theories \(PA'=PA\) \(+\) local reflection principle and \(PA_{\omega}=\cup_{n\in \omega}PA_ n\) are relatively interpretable. The result seems interesting because \(PA_{\omega}\subset PA'\) and \(PA_{\omega}\) is in the following sense much weaker than PA': \(PA_{\omega}\) is contained in some finite extension of PA and PA' is not.
Mathematics Subject Classification ID: 03F25
Mathematics Subject Classification ID: 03F07
Mathematics Subject Classification ID: 03B45
zbMATH DE Number: 4027458
zbMATH Keywords: provability
zbMATH Keywords: modal logic
zbMATH Keywords: interpretability of extensions of arithmetic
zbMATH Keywords: local reflection principle
reviewed by: Sergei Artemov
MaRDI profile type: Publication
cites work: One hundred and two problems in mathematical logic
cites work: Arithmetization of metamathematics in a general setting
cites work: Provability interpretations of modal logic
cites work: Q4068699
cites work: Q3964565
description: scientific article; zbMATH DE number 4027458